Determining The Range Of The Function F(x) = (1/7)(9)^x
In the realm of mathematics, understanding the range of a function is crucial for comprehending its behavior and potential applications. The range represents the set of all possible output values that a function can produce. For the given function, f(x) = (1/7)(9)^x, we embark on a journey to determine its range, exploring the fundamental properties of exponential functions and their impact on the output values.
Grasping the Essence of Exponential Functions
To effectively determine the range of f(x) = (1/7)(9)^x, we must first grasp the essence of exponential functions. In general, an exponential function takes the form f(x) = a * b^x, where a is a constant coefficient and b is the base. The base b plays a pivotal role in shaping the function's behavior. When b is greater than 1, as in our case with b = 9, the function exhibits exponential growth, meaning that as x increases, the function's output values increase rapidly. Conversely, when b is between 0 and 1, the function demonstrates exponential decay, where the output values decrease as x increases. The coefficient a acts as a vertical stretch or compression factor, influencing the overall scale of the function's output values.
Exponential Growth in Action
In our function, f(x) = (1/7)(9)^x, the base b = 9 is greater than 1, signifying exponential growth. As x takes on larger positive values, the term 9^x grows exponentially, causing the overall function value to increase rapidly. The coefficient (1/7) scales down the output values, but the fundamental exponential growth pattern remains intact. Conversely, as x takes on larger negative values, the term 9^x approaches 0, but it never actually reaches 0. This is a crucial characteristic of exponential functions: they approach 0 asymptotically as x tends towards negative infinity.
The Significance of the Coefficient
The coefficient (1/7) in our function plays a vital role in determining the range. Since (1/7) is a positive number, it ensures that the output values of the function will always be positive. If the coefficient were negative, the output values would be negative. If the coefficient were 0, the function would simply be a horizontal line at y = 0. Therefore, the positive coefficient (1/7) restricts the range to positive values.
Delving into the Range Determination
With a firm grasp of exponential functions, we can now delve into determining the range of f(x) = (1/7)(9)^x. As we discussed earlier, the exponential term 9^x is always positive, regardless of the value of x. Multiplying this positive term by the positive coefficient (1/7) ensures that the output values of the function will always be positive. Furthermore, as x approaches negative infinity, the term 9^x approaches 0, but it never actually reaches 0. Consequently, the function f(x) = (1/7)(9)^x approaches 0, but it never actually reaches 0. On the other hand, as x approaches positive infinity, the term 9^x grows without bound, causing the function f(x) = (1/7)(9)^x to also grow without bound.
Constructing the Range
Based on these observations, we can construct the range of the function f(x) = (1/7)(9)^x. The function can take on any positive value, but it can never be 0 or negative. Therefore, the range of the function is the set of all real numbers greater than 0. In interval notation, we can express this range as (0, ∞). This signifies that the function's output values can be any number between 0 and infinity, excluding 0 itself.
Formalizing the Range with Mathematical Notation
To express the range of the function f(x) = (1/7)(9)^x formally using mathematical notation, we can write:
Range of f(x) = {y ∈ ℝ | y > 0}
This notation reads as follows: "The range of f(x) is the set of all y that belong to the set of real numbers (ℝ) such that y is greater than 0." This notation concisely captures the essence of the range, emphasizing that the output values are positive real numbers.
Conclusion: The Range of Exponential Growth
In conclusion, the range of the exponential function f(x) = (1/7)(9)^x is the set of all real numbers greater than 0, expressed as (0, ∞) in interval notation or {y ∈ ℝ | y > 0} in mathematical notation. This range reflects the fundamental behavior of exponential growth, where the function's output values increase rapidly as the input increases, always remaining positive due to the positive coefficient and the nature of the exponential term. Understanding the range of a function is crucial for comprehending its overall behavior and potential applications, and this exploration of f(x) = (1/7)(9)^x has provided valuable insights into the characteristics of exponential functions.
Determining the range of a function is a fundamental concept in mathematics, allowing us to understand the set of all possible output values a function can produce. In this article, we will delve into finding the range of the function f(x) = (1/7)(9)^x, a classic example of an exponential function. We will explore the properties of exponential functions, analyze the given function's behavior, and ultimately define its range.
Understanding Exponential Functions
Before we can determine the range of f(x) = (1/7)(9)^x, it's essential to understand the nature of exponential functions. A general exponential function is expressed as f(x) = a * b^x, where:
- a is a constant coefficient.
- b is the base (a positive real number not equal to 1).
- x is the exponent.
The base b dictates the function's behavior: when b > 1, the function exhibits exponential growth, and when 0 < b < 1, it exhibits exponential decay. The coefficient a scales the function vertically.
Key Properties of Exponential Functions
- Domain: Exponential functions are defined for all real numbers, meaning the domain is (-∞, ∞).
- Asymptote: When b > 1, as x approaches -∞, f(x) approaches 0 but never reaches it. This creates a horizontal asymptote at y = 0. When 0 < b < 1, as x approaches ∞, f(x) approaches 0.
- Range: The range depends on the coefficient a. If a > 0, the range is (0, ∞). If a < 0, the range is (-∞, 0).
Analyzing f(x) = (1/7)(9)^x
Now, let's analyze our specific function, f(x) = (1/7)(9)^x. Here, a = 1/7 and b = 9. Since b = 9 > 1, this function exhibits exponential growth. The coefficient a = 1/7 is positive, which will influence the range.
Deconstructing the Function
To fully understand the function's behavior, let's break it down:
- 9^x: This is the core exponential part. As x increases, 9^x increases rapidly. As x decreases towards negative infinity, 9^x approaches 0 but never becomes 0.
- (1/7)(9)^x: Multiplying 9^x by (1/7) compresses the function vertically, but it doesn't change the fundamental exponential growth pattern. The function will still approach 0 as x approaches negative infinity and grow rapidly as x increases.
Determining the Range of f(x)
With our understanding of exponential functions and the specific behavior of f(x) = (1/7)(9)^x, we can now determine its range.
- Positive Values: Since 9^x is always positive for any real number x, and (1/7) is a positive coefficient, the product (1/7)(9)^x will always be positive.
- Approaching Zero: As x approaches -∞, 9^x approaches 0. Consequently, (1/7)(9)^x also approaches 0, but it never actually reaches 0.
- Unbounded Growth: As x approaches ∞, 9^x grows without bound, and so does (1/7)(9)^x.
The Range in Interval Notation
Based on these observations, the range of f(x) = (1/7)(9)^x is all positive real numbers. In interval notation, this is expressed as (0, ∞).
Mathematical Notation
We can also express the range using set notation:
Range of f(x) = {y ∈ ℝ | y > 0}
This reads as "the range of f(x) is the set of all y that belong to the real numbers (ℝ) such that y is greater than 0."
Graphical Representation
A graph of f(x) = (1/7)(9)^x visually confirms our findings. The function starts very close to the x-axis (but never touches it) and increases exponentially as x increases. The graph lies entirely above the x-axis, indicating that the range consists only of positive values.
Practical Implications
Understanding the range of a function is not just a theoretical exercise. It has practical implications in various fields, such as:
- Modeling Growth: Exponential functions are used to model population growth, compound interest, and other phenomena where quantities increase rapidly. Knowing the range helps us understand the possible outcomes.
- Physics and Engineering: Exponential functions appear in equations describing radioactive decay, electrical circuits, and other physical systems. The range provides information about the possible values of these physical quantities.
Conclusion: Defining the Reach of Exponential Functions
In this article, we have thoroughly explored the range of the function f(x) = (1/7)(9)^x. By understanding the properties of exponential functions, analyzing the specific function's behavior, and considering its graphical representation, we have definitively determined that the range is (0, ∞). This means the function can take on any positive value but never zero or a negative value. This understanding is crucial for applying exponential functions in various real-world scenarios and further mathematical explorations.
In mathematics, the range of a function is a critical concept, representing the set of all possible output values the function can produce. For the function f(x) = (1/7)(9)^x, we aim to identify its range through a systematic analysis of exponential functions and their properties. This article will guide you through the process of determining the range of f(x), providing a clear understanding of the function's behavior and its output values.
Foundational Concepts: Exponential Functions Revisited
To effectively determine the range of f(x) = (1/7)(9)^x, it's essential to revisit the fundamental properties of exponential functions. Generally, an exponential function is defined as f(x) = a * b^x, where:
- a represents a constant coefficient.
- b is the base, a positive real number not equal to 1.
- x denotes the exponent, which can be any real number.
The base, b, plays a crucial role in shaping the function's behavior. When b > 1, the function exhibits exponential growth, meaning the output values increase rapidly as x increases. Conversely, when 0 < b < 1, the function demonstrates exponential decay, where the output values decrease as x increases. The coefficient a acts as a vertical scaling factor, influencing the magnitude of the output values.
Key Characteristics of Exponential Functions
- Domain: Exponential functions are defined for all real numbers, thus, the domain is (-∞, ∞).
- Asymptotic Behavior: When b > 1, as x approaches negative infinity (-∞), f(x) approaches 0 but never reaches it. This creates a horizontal asymptote at y = 0. When 0 < b < 1, as x approaches positive infinity (∞), f(x) approaches 0.
- Range and the Coefficient: The range of the exponential function is heavily influenced by the coefficient a. If a > 0, the range is (0, ∞), indicating all positive real numbers. If a < 0, the range is (-∞, 0), encompassing all negative real numbers.
Analyzing the Function f(x) = (1/7)(9)^x
Now, let's focus on our specific function, f(x) = (1/7)(9)^x. Here, we can identify a = 1/7 and b = 9. Given that b = 9 > 1, this function demonstrates exponential growth. Furthermore, the coefficient a = 1/7 is positive, which will have a significant impact on the function's range.
Deconstructing the Components
To gain a comprehensive understanding of the function's behavior, let's break it down into its components:
- 9^x: This core exponential term is always positive for any real number x. As x increases, 9^x increases exponentially. As x approaches negative infinity, 9^x approaches 0 but never equals 0.
- (1/7)(9)^x: Multiplying 9^x by (1/7) results in a vertical compression of the function, but it doesn't alter the fundamental exponential growth pattern. The function will still approach 0 as x approaches negative infinity and grow exponentially as x increases.
Step-by-Step Determination of the Range
With a solid understanding of exponential functions and the specific behavior of f(x) = (1/7)(9)^x, we can now systematically determine its range.
Step 1: Positive Values
Since 9^x is always positive for any real number x, and (1/7) is a positive coefficient, the product (1/7)(9)^x will always yield a positive value. This is a crucial observation that restricts the range to positive numbers.
Step 2: Asymptotic Behavior
As x approaches negative infinity (-∞), 9^x approaches 0. Consequently, (1/7)(9)^x also approaches 0. However, it's essential to note that the function never actually reaches 0. This asymptotic behavior defines the lower bound of the range.
Step 3: Unbounded Growth
As x approaches positive infinity (∞), 9^x grows without bound, and thus, (1/7)(9)^x also grows without bound. This indicates that there is no upper limit to the function's output values.
Step 4: Defining the Range
Based on these observations, we can conclude that the range of f(x) = (1/7)(9)^x is all positive real numbers. The function can take on any positive value, but it cannot be zero or negative.
Expressing the Range: Interval and Set Notations
To formally express the range of f(x) = (1/7)(9)^x, we can use interval notation and set notation.
Interval Notation
In interval notation, the range is expressed as (0, ∞). The parentheses indicate that the endpoints are not included in the range, signifying that the function can approach 0 but never reach it.
Set Notation
Alternatively, we can express the range using set notation:
Range of f(x) = {y ∈ ℝ | y > 0}
This notation reads as "the range of f(x) is the set of all y that belong to the set of real numbers (ℝ) such that y is greater than 0."
Visual Confirmation: Graphical Representation
A graph of f(x) = (1/7)(9)^x provides a visual confirmation of our findings. The function's curve lies entirely above the x-axis, indicating that the output values are always positive. The curve approaches the x-axis as x decreases but never intersects it, confirming the lower bound of the range. As x increases, the curve rises exponentially, demonstrating the unbounded growth.
Real-World Applications: The Significance of Range
Understanding the range of a function is not merely an abstract mathematical exercise. It has practical implications in various real-world applications.
- Growth Models: Exponential functions are commonly used to model phenomena such as population growth and compound interest. The range helps us understand the potential outcomes of these growth processes.
- Physical Sciences: Exponential functions appear in equations describing radioactive decay, electrical circuits, and other physical systems. The range provides insights into the possible values of physical quantities.
Conclusion: Unveiling the Range of Exponential Growth
In this comprehensive analysis, we have successfully determined the range of the exponential function f(x) = (1/7)(9)^x. By revisiting the fundamental properties of exponential functions, dissecting the function's components, and employing a step-by-step approach, we have established that the range is (0, ∞), encompassing all positive real numbers. This understanding is crucial for applying exponential functions in diverse contexts and for further mathematical explorations.